Mahāvīra (mathematician)

Template:Short description Template:Use dmy dates Template:Use Indian English Template:Infobox religious biography Mahāvīra (or Mahaviracharya, "Mahavira the Teacher") was a 9th-century Indian Jain mathematician possibly born in Mysore, in India.Template:SfnTemplate:SfnTemplate:Sfn He authored Gaṇita-sāra-saṅgraha (Ganita Sara Sangraha) or the Compendium on the gist of Mathematics in 850 CE.Template:Sfn He was patronised by the Rashtrakuta emperor Amoghavarsha.Template:Sfn He separated astrology from mathematics. It is the earliest Indian text entirely devoted to mathematics.<ref>The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the ... by Clifford A. Pickover: page 88</ref> He expounded on the same subjects on which Aryabhata and Brahmagupta contended, but he expressed them more clearly. His work is a highly syncopated approach to algebra and the emphasis in much of his text is on developing the techniques necessary to solve algebraic problems.<ref>Algebra: Sets, Symbols, and the Language of Thought by John Tabak: p.43</ref> He is highly respected among Indian mathematicians, because of his establishment of terminology for concepts such as equilateral, and isosceles triangle; rhombus; circle and semicircle.<ref>Geometry in Ancient and Medieval India by T. A. Sarasvati Amma: page 122</ref> Mahāvīra's eminence spread throughout southern India and his books proved inspirational to other mathematicians in Southern India.Template:Sfn It was translated into the Telugu language by Pavuluri Mallana as Saara Sangraha Ganitamu.<ref>Census of the Exact Sciences in Sanskrit by David Pingree: page 388</ref>

He discovered algebraic identities like a³ = a (a + b) (a − b) + b² (a − b) + b³.Template:Sfn He also found out the formula for ⁿC_r as
[n (n − 1) (n − 2) ... (n − r + 1)] / [r (r − 1) (r − 2) ... 2 * 1].Template:Sfn He devised a formula which approximated the area and perimeters of ellipses and found methods to calculate the square of a number and cube roots of a number.Template:Sfn He asserted that the square root of a negative number does not exist.Template:Sfn Arithmetic operations utilized in his works like Gaṇita-sāra-saṅgraha(Ganita Sara Sangraha) uses decimal place-value system and include the use of zero. However, he erroneously states that a number divided by zero remains unchanged.<ref>Template:Cite book</ref>

Rules for decomposing fractions

Mahāvīra's Gaṇita-sāra-saṅgraha gave systematic rules for expressing a fraction as the sum of unit fractions.<ref name=k497>Template:Harvnb</ref> This follows the use of unit fractions in Indian mathematics in the Vedic period, and the Śulba Sūtras' giving an approximation of Template:Radic equivalent to <math>1 + \tfrac13 + \tfrac1{3\cdot4} - \tfrac1{3\cdot4\cdot34}</math>.<ref name=k497/>

In the Gaṇita-sāra-saṅgraha (GSS), the second section of the chapter on arithmetic is named kalā-savarṇa-vyavahāra (lit. "the operation of the reduction of fractions"). In this, the bhāgajāti section (verses 55–98) gives rules for the following:<ref name=k497/>

• To express 1 as the sum of n unit fractions (GSS kalāsavarṇa 75, examples in 76):<ref name=k497/>

rūpāṃśakarāśīnāṃ rūpādyās triguṇitā harāḥ kramaśaḥ /

dvidvitryaṃśābhyastāv ādimacaramau phale rūpe //{{#if:|

|}}{{#if:|

— {{#if:|, in }}Template:Comma separated entries

}}{{#invoke:Check for unknown parameters|check|unknown=Template:Main other|preview=Page using Template:Blockquote with unknown parameter "_VALUE_"|ignoreblank=y| 1 | 2 | 3 | 4 | 5 | author | by | char | character | cite | class | content | multiline | personquoted | publication | quote | quotesource | quotetext | sign | source | style | text | title | ts }}

When the result is one, the denominators of the quantities having one as numerators are [the numbers] beginning with one and multiplied by three, in order. The first and the last are multiplied by two and two-thirds [respectively].{{#if:|

|}}{{#if:|

— {{#if:|, in }}Template:Comma separated entries

}}{{#invoke:Check for unknown parameters|check|unknown=Template:Main other|preview=Page using Template:Blockquote with unknown parameter "_VALUE_"|ignoreblank=y| 1 | 2 | 3 | 4 | 5 | author | by | char | character | cite | class | content | multiline | personquoted | publication | quote | quotesource | quotetext | sign | source | style | text | title | ts }}

<math> 1 = \frac1{1 \cdot 2} + \frac1{3} + \frac1{3^2} + \dots + \frac1{3^{n-2}} + \frac1{\frac23 \cdot 3^{n-1}} </math>

• To express 1 as the sum of an odd number of unit fractions (GSS kalāsavarṇa 77):<ref name=k497/>

<math>1 = \frac1{2\cdot 3 \cdot 1/2} + \frac1{3 \cdot 4 \cdot 1/2} + \dots + \frac1{(2n-1) \cdot 2n \cdot 1/2} + \frac1{2n \cdot 1/2} </math>

• To express a unit fraction <math>1/q</math> as the sum of n other fractions with given numerators <math>a_1, a_2, \dots, a_n</math> (GSS kalāsavarṇa 78, examples in 79):

<math>\frac1q = \frac{a_1}{q(q+a_1)} + \frac{a_2}{(q+a_1)(q+a_1+a_2)} + \dots + \frac{a_{n-1}}{(q+a_1+\dots+a_{n-2})(q+a_1+\dots+a_{n-1})} + \frac{a_n}{a_n(q+a_1+\dots+a_{n-1})}</math>

• To express any fraction <math>p/q</math> as a sum of unit fractions (GSS kalāsavarṇa 80, examples in 81):<ref name=k497/>

Choose an integer i such that <math>\tfrac{q+i}{p}</math> is an integer r, then write

<math> \frac{p}{q} = \frac{1}{r} + \frac{i}{r \cdot q} </math>

and repeat the process for the second term, recursively. (Note that if i is always chosen to be the smallest such integer, this is identical to the greedy algorithm for Egyptian fractions.)

• To express a unit fraction as the sum of two other unit fractions (GSS kalāsavarṇa 85, example in 86):<ref name=k497/>

<math>\frac1{n} = \frac1{p\cdot n} + \frac1{\frac{p\cdot n}{n-1}}</math> where <math>p</math> is to be chosen such that <math>\frac{p\cdot n}{n-1}</math> is an integer (for which <math>p</math> must be a multiple of <math>n-1</math>).

<math>\frac1{a\cdot b} = \frac1{a(a+b)} + \frac1{b(a+b)}</math>

• To express a fraction <math>p/q</math> as the sum of two other fractions with given numerators <math>a</math> and <math>b</math> (GSS kalāsavarṇa 87, example in 88):<ref name=k497/>

<math>\frac{p}{q} = \frac{a}{\frac{ai+b}{p}\cdot\frac{q}{i}} + \frac{b}{\frac{ai+b}{p} \cdot \frac{q}{i} \cdot{i}}</math> where <math>i</math> is to be chosen such that <math>p</math> divides <math>ai + b</math>

Some further rules were given in the Gaṇita-kaumudi of Nārāyaṇa in the 14th century.<ref name=k497/>

See also

• List of Indian mathematicians

References

• Bibhutibhusan Datta and Avadhesh Narayan Singh (1962). History of Hindu Mathematics: A Source Book.
• Template:DSB (Available, along with many other entries from other encyclopaedias for other Mahāvīra-s, online.)
• Template:Citation
• Template:Citation
• Template:MacTutor Biography
• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation